Suppose now that \(\phi \)  is smooth. For \(x\in X^0\) , we can restricts to a \(\mathbb {C}^\ast \) -invariant étale (resp. open in the analytic topology) neighborhood of \(\phi (x)\)  in which \(Y\)  is affine, and further restricts to a \(\mathbb {C}^\ast \) -invariant étale (resp. open in the analytic topology) neighborhood of \(x\)  in \(X\)  which is affine. As argued in the proof of proposition 2.43 of [1]}, because \(\mathbb {C}^\ast \)  is a torus, an affine \(\mathbb {C}^\ast \) -equivariant space can be written as \(Spec(R[x_1,...,x_n]/(h_1,...,h_r))\)  where the \(x_i\)  are \(\mathbb {C}^\ast \) -equivariant coordinates, and the \(h_i\)  are \(\mathbb {C}^\ast \) -equivariant polynomial. Then the \(\mathbb {C}^\ast \) -equivariant smooth map \(\phi :X\rightarrow Y\)  is étale locally given by the dual of the map of rings:
\(\tilde{R}[x_1,...,x_n]/(h_1,...,h_r)\rightarrow \tilde{R}[y_1,...,y_m]/(k_1,...,k_s)\) 
